2.
(a) Find an equation of the tangent plane to the surface
x4 +y4 +z4 = 18 at (2, 1, 1). Find a
derivative in direction (2,2,1) at point (2,1,1). (b) Use Lagrange
multipliers to find the minimum and maximum values of f(x,y,z) = 8x
+ y + z on the surface x4 + y4 + z4 = 18.
Find the equation of the tangent plane to the surface,
(4x^2)(y^3) + (5yz) + (2xz^3) = 7
at the point P(-1,1,1). Also nd the parametric equation of the
normal
line to that surface at that point . Sketch a picture that
illustrates what this
is all about.
1a.Find the equation of the tangent plane to the surface √ x +
√y + √ z = 4 at P(1, 1, 4).
1b.Let f be a function of x and y such that fx = 3x − 5y and fy
= 2y − 5x, which of the following is always TRUE?
a. (0, 0) is not a critical point of f.
b. f has a local minimum at (0, 0)
c. f has a local maximum at (0, 0)...
a. Find the tangent plane to z=(10-x^2-2y^2)^2 at (1,2,1).
b. Find the equation of the tangent plane to the level surface
for w=x^2y+y^2z+xyz^2 when w=8 at (1,2,1).
Find an equation of the tangent plane to the surface
x5+5z2ey−x=848 at point P=(3, 4, 11e√).
(Use symbolic notation and fractions where needed. Your answer
should be in the form ax+by+cz=1.)
1. Find an equation for the line in the xy−plane that is tangent
to the curve at the point corresponding to the given value of t.
Also, find the value of d^2y/dx^2 at this point. x=sec t, y=tan t,
t=π/6
2. Find the length of the parametric curve: x=cos t, y=t+sin t,
0 ≤ t ≤ π. Hint:To integrate , use the identity, and
complete the integral.
Find the equation of the tangent plane and the
parametric equations for the normal line to the surface
x2 + y2 - z = 0 at the point P(4,-1, 6).
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